126 research outputs found
Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS
We consider the 2-dimensional focusing mass critical NLS with an
inhomogeneous nonlinearity: . From
standard argument, there exists a threshold such that solutions
with are global in time while a finite time blow up
singularity formation may occur for . In this paper, we
consider the dynamics at threshold and give a necessary and
sufficient condition on to ensure the existence of critical mass finite
time blow up elements. Moreover, we give a complete classification in the
energy class of the minimal finite time blow up elements at a non degenerate
point, hence extending the pioneering work by Merle who treated the pseudo
conformal invariant case
Blow up for the critical gKdV equation III: exotic regimes
We consider the blow up problem in the energy space for the critical (gKdV)
equation in the continuation of part I and part II.
We know from part I that the unique and stable blow up rate for solutions
close to the solitons with strong decay on the right is . In this paper,
we construct non-generic blow up regimes in the energy space by considering
initial data with explicit slow decay on the right in space. We obtain finite
time blow up solutions with speed where as well as
global in time growing up solutions with both exponential growth or power
growth. These solutions can be taken with initial data arbitrarily close to the
ground state solitary wave
Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5
We consider the semilinear wave equation with focusing energy-critical
nonlinearity in space dimension 5 with radial data. It is known that a solution
which blows up at in a neighborhood (in the energy
norm) of the family of solitons , asymptotically decomposes in the
energy space as a sum of a bubble and an asymptotic profile
, where and . We construct a blow-up solution of this type such that
is any pair of sufficiently regular functions with . For these solutions the concentration rate is . We
also provide examples of solutions with concentration rate for , related to the behaviour of the asymptotic profile
near the origin.Comment: 39 pages; the new version takes into account the remarks of the
referee
Blow up for the critical gKdV equation II: minimal mass dynamics
We fully revisit the near soliton dynamics for the mass critical (gKdV)
equation.
In Part I, for a class of initial data close to the soliton, we prove that
only three scenario can occur:
(BLOW UP) the solution blows up in finite time in a universal regime with
speed ;
(SOLITON) the solution is global and converges to a soliton in large time;
(EXIT) the solution leaves any small neighborhood of the modulated family of
solitons in the scale invariant norm.
Regimes (BLOW UP) and (EXIT) are proved to be stable. We also show in this
class that any nonpositive energy initial data (except solitons) yields finite
time blow up, thus obtaining the classification of the solitary wave at zero
energy.
In Part II, we classify minimal mass blow up by proving existence and
uniqueness (up to invariances of the equation) of a minimal mass blow up
solution . We also completely describe the blow up behavior of .
Second, we prove that is the universal attractor in the (EXIT) case,
i.e. any solution as above in the (EXIT) case is close to (up to
invariances) in at the exit time. In particular, assuming scattering for
(in large positive time), we obtain that any solution in the (EXIT)
scenario also scatters, thus achieving the description of the near soliton
dynamics
LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I
CONTENTS
J. Bona
Derivation and some fundamental properties of nonlinear dispersive waves equations
F. Planchon
Schr\"odinger equations with variable coecients
P. Rapha\"el
On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio
Blow up dynamics for smooth equivariant solutions to the energy critical Schr\"odinger map
We consider the energy critical Schr\"odinger map problem with the 2-sphere
target for equivariant initial data of homotopy index . We show the
existence of a codimension one set of smooth well localized initial data
arbitrarily close to the ground state harmonic map in the energy critical norm,
which generates finite time blow up solutions. We give a sharp description of
the corresponding singularity formation which occurs by concentration of a
universal bubble of energy
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